Question 447772
You are over-thinking the problem, and with that solution, you are likely to make a mistake, since you cannot directly assume you can square both sides with the absolute value signs.


Just note that you have a sum of two absolute value expressions, and the absolute value of any quantity is nonnegative. |2x+1| > 0 unless x = -1/2, and |x-4| > 0 unless x = 4. Since x cannot be -1/2 and 4 at the same time, at least one of the absolute value expressions must be greater than 0, so the inequality holds for all real numbers x.